Levitt–McCammond's finiteness conjecture for thoroughly crumpled planes

Let $X$ be a compact locally CAT(0) triangle-square complex. A flat is thoroughly crumpled if it is crumpled and there is a bound on the number of cells in each region. Levitt–McCammond's finiteness conjecture. There are only finitely many distinct thoroughly crumpled planes, up to cellular isometries, that embed in $\widetilde{X}$. This is the second original Levitt–McCammond conjecture relevant to the proposed biautomaticity argument; the supplied text does not establish its resolution.